When are tris used effectively?
In terms of quality I suggest you to model in quads. The simplest reason is:


  The flexibility to soften and sharpen edges is key to build high quality
  models.




Example geometry with a soft edge on the left side and more sharp edge on the right  

The best way to achieve this, is to build consistent topology to be fast in selecting  Alt+Shift+RMB and creating edge loops Ctrl+R, which depends on quads.



Create edge loop issue 

For the picture of your question, I assume that automatic algorithms (e.g. bevel modifiers) have been applied to improve the edges. A side effect of such algorithms is that unwanted edges appear (triangles at the corners), but I can't really see if there is a reason for this. 


  For sure there are exceptions to use triangles.


In terms of speed artists often use modifiers on simple geometry to build complex shapes - side effect can be a geometry of triangles. For the crumpled paper I could also imagine that here some kind of modifier on a icosphere was used.

But it really depends on what do you want to achieve.